One way to see this is to note that the graph of the functionf(x) = x2 is a parabola whose vertex is at the origin (0, 0). Therefore, the graph of the function f(x − h) = (x − h)2 is a parabola shifted to the right by h whose vertex is at (h, 0), as shown in the top figure. In contrast, the graph of the function f(x) + k = x2 + k is a parabola shifted upward by k whose vertex is at (0, k), as shown in the center figure. Combining both horizontal and vertical shifts yields f(x − h) + k = (x − h)2 + k is a parabola shifted to the right by h and upward by k whose vertex is at (h, k), as shown in the bottom figure.
Solving quadratic equations
Completing the square may be used to solve any quadratic equation. For example:
The first step is to complete the square:
Next we solve for the squared term:
Then either
This can be applied to any quadratic equation. When the x2 has a coefficient other than 1, the first step is to divide out the equation by this coefficient: for an example see the non-monic case below.
Irrational and complex roots
Unlike methods involving factoring the equation, which is reliable only if the roots are rational, completing the square will find the roots of a quadratic equation even when those roots are irrational or complex. For example, consider the equation
Completing the square gives
In terser language:
Equations with complex roots can be handled in the same way. For example:
Non-monic case
For an equation involving a non-monic quadratic, the first step to solving them is to divide through by the coefficient of x2. For example:
Applying this procedure to the general form of a quadratic equation leads to the quadratic formula.
Other applications
Integration
Completing the square may be used to evaluate any integral of the form
For example, consider the integral
Completing the square in the denominator gives:
This can now be evaluated by using the substitutionu = x + 3, which yields
A matrixM is idempotent when M2 = M. Idempotent matrices generalize the idempotent properties of 0 and 1. The completion of the square method of addressing the equation
The matrix will be idempotent provided which, upon completing the square, becomes
ab
Geometric perspective
Consider completing the square for the equation
Since x2 represents the area of a square with side of length x, and bx represents the area of a rectangle with sides b and x, the process of completing the square can be viewed as visual manipulation of rectangles.
Simple attempts to combine the x2 and the bx rectangles into a larger square result in a missing corner. The term (b/2)2 added to each side of the above equation is precisely the area of the missing corner, whence derives the terminology "completing the square".[7]
A variation on the technique
As conventionally taught, completing the square consists of adding the third term, v2 to
uvuv
Example: the sum of a positive number and its reciprocal
By writing
xx
Example: factoring a simple quartic polynomial
Consider the problem of factoring the polynomial
This is
x2x2
The same argument shows that is always factorizable as
"Completing the square" consists to remark that the two first terms of a quadratic polynomial are also the first terms of the square of a linear polynomial, and to use this for expressing the quadratic polynomial as the sum of a square and a constant.
Completing the cube is a similar technique that allows to transform a cubic polynomial into a cubic polynomial without term of degree two.
More precisely, if
is a polynomial in x such that its two first terms are the two first terms of the expanded form of
provides a cubic polynomial in without term of degree two, which is called the depressed form of the original polynomial.
This transformation is generally the first step of the mehods for solving the general cubic equation.
More generally, a similar transformation can be used for removing terms of degree in polynomials of degree , which is called Tschirnhaus transformation.
References
^Dionissios T. Hristopulos (2020). Random Fields for Spatial Data Modeling: A Primer for Scientists and Engineers. Springer Nature. p. 267. ISBN 978-94-024-1918-4. Extract of page 267
^James R. Brannan; William E. Boyce (2015). Differential Equations: An Introduction to Modern Methods and Applications (3rd ed.). John Wiley & Sons. p. 314. ISBN 978-1-118-98122-1. Extract of page 314
^Stephen L. Campbell; Richard Haberman (2011). Introduction to Differential Equations with Dynamical Systems (illustrated ed.). Princeton University Press. p. 214. ISBN 978-1-4008-4132-5. Extract of page 214
^Tony Philips, "Completing the Square", American Mathematical Society Feature Column, 2020.
^Hughes, Barnabas. "Completing the Square - Quadratics Using Addition". Math Association of America. Retrieved 2022-10-21.
^Narasimhan, Revathi (2008). Precalculus: Building Concepts and Connections. Cengage Learning. pp. 133–134. ISBN 978-0-618-41301-0., Section Formula for the Vertex of a Quadratic Function, page 133–134, figure 2.4.8
^Carroll, Maureen T.; Rykken, Elyn (2018). Geometry: The Line and the Circle. AMS/MAA Textbooks. American Mathematical Society. p. 162. ISBN 978-1-4704-4843-1. Retrieved 2024-03-31.
Algebra 1, Glencoe, ISBN 0-07-825083-8, pages 539–544
Algebra 2, Saxon, ISBN 0-939798-62-X, pages 214–214, 241–242, 256–257, 398–401
External links
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